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Difference between revisions of "1997 AIME Problems/Problem 14"

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We need <math>\cos (\frac{2\pi m}{1997})\cos (\frac{2\pi n}{1997}) + \sin (\frac{2\pi m}{1997})\sin (\frac{2\pi n}{1997}) \ge \frac{\sqrt{3}}{2}</math>. The [[Trigonometric identities|cosine difference identity]] simplifies that to <math>\cos\left(\frac{2\pi m}{1997} - \frac{2\pi n}{1997}\right) \ge \frac{\sqrt{3}}{2}</math>. Thus, <math>|m - n| \le \frac{\pi}{6} \cdot 2 \cdot \frac{1997}{2 \pi} = \frac{1997}{6}=166</math>.
 
We need <math>\cos (\frac{2\pi m}{1997})\cos (\frac{2\pi n}{1997}) + \sin (\frac{2\pi m}{1997})\sin (\frac{2\pi n}{1997}) \ge \frac{\sqrt{3}}{2}</math>. The [[Trigonometric identities|cosine difference identity]] simplifies that to <math>\cos\left(\frac{2\pi m}{1997} - \frac{2\pi n}{1997}\right) \ge \frac{\sqrt{3}}{2}</math>. Thus, <math>|m - n| \le \frac{\pi}{6} \cdot 2 \cdot \frac{1997}{2 \pi} = \frac{1997}{6}=166</math>.
  
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Therefore, <math>\displaystyle m</math> and <math>\displaystyle n</math> cannot be more than <math>\displaystyle 166</math> away from each other.  This means that for a given value of <math>\displaystyle m</math>, there are <math>\displastyle 332</math> values for <math>\displaystyle n</math> that satisfy the inequality: <math>\displastyle 166</math> of them are greater than <math>\displastyle m</math>, and <math>\displaystyle 166</math> are less than <math>\displaystyle m</math>.  Since <math>\displastyle m</math> and <math>\displaystyle n</math> must be distinct, <math>\displaystyle n</math> can have <math>\displastyle 1996</math> possible values.  Therefore, the probability is <math>\displaystyle\frac{332}{1996}=\frac{83}{499}</math>.  The answer is then <math>\displaystyle 499+83=582</math>
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Therefore, <math>\displaystyle m</math> and <math>\displaystyle n</math> cannot be more than <math>\displaystyle 166</math> away from each other.  This means that for a given value of <math>\displaystyle m</math>, there are <math>\displaystyle 332</math> values for <math>\displaystyle n</math> that satisfy the inequality: <math>\displaystyle 166</math> of them are greater than <math>\displaystyle m</math>, and <math>\displaystyle 166</math> are less than <math>\displaystyle m</math>.  Since <math>\displaystyle m</math> and <math>\displaystyle n</math> must be distinct, <math>\displaystyle n</math> can have <math>\displaystyle 1996</math> possible values.  Therefore, the probability is <math>\displaystyle\frac{332}{1996}=\frac{83}{499}</math>.  The answer is then <math>\displaystyle 499+83=582</math>
  
 
== See also ==
 
== See also ==

Revision as of 09:43, 8 March 2007

Problem

Let $\displaystyle v$ and $\displaystyle w$ be distinct, randomly chosen roots of the equation $\displaystyle z^{1997}-1=0$. Let $\displaystyle \frac{m}{n}$ be the probability that $\displaystyle\sqrt{2+\sqrt{3}}\le\left|v+w\right|$, where $\displaystyle m$ and $\displaystyle n$ are relatively prime positive integers. Find $\displaystyle m+n$.

Solution

$\displaystyle z^{1997}=1$

By De Moivre's Theorem, we find that

$\displaystyle z=\cos\left(\frac{2\pi k}{1997}\right)+i\sin\left(\frac{2\pi k}{1997}\right)$

Now, let $\displaystyle v$ be the root corresponding to $\displaystyle \theta=\frac{2\pi m}{1997}$, and let $\displaystyle w$ be the root corresponding to $\displaystyle \theta=\frac{2\pi n}{1997}$. The magnitude of $\displaystyle v+w$ is therefore:

$\sqrt{\left(\cos\left(\frac{2\pi m}{1997}\right) + \cos\left(\frac{2\pi n}{1997}\right)\right)^2 + \left(\sin\left(\frac{2\pi m}{1997}\right) + \sin\left(\frac{2\pi n}{1997}\right)\right)^2}$
$=\sqrt{2 + 2\cos\left(\frac{2\pi m}{1997}\right)\cos\left(\frac{2\pi n}{1997}\right) + 2\sin\left(\frac{2\pi m}{1997}\right)\sin\left(\frac{2\pi n}{1997}\right)}$

We need $\cos (\frac{2\pi m}{1997})\cos (\frac{2\pi n}{1997}) + \sin (\frac{2\pi m}{1997})\sin (\frac{2\pi n}{1997}) \ge \frac{\sqrt{3}}{2}$. The cosine difference identity simplifies that to $\cos\left(\frac{2\pi m}{1997} - \frac{2\pi n}{1997}\right) \ge \frac{\sqrt{3}}{2}$. Thus, $|m - n| \le \frac{\pi}{6} \cdot 2 \cdot \frac{1997}{2 \pi} = \frac{1997}{6}=166$.

Therefore, $\displaystyle m$ and $\displaystyle n$ cannot be more than $\displaystyle 166$ away from each other. This means that for a given value of $\displaystyle m$, there are $\displaystyle 332$ values for $\displaystyle n$ that satisfy the inequality: $\displaystyle 166$ of them are greater than $\displaystyle m$, and $\displaystyle 166$ are less than $\displaystyle m$. Since $\displaystyle m$ and $\displaystyle n$ must be distinct, $\displaystyle n$ can have $\displaystyle 1996$ possible values. Therefore, the probability is $\displaystyle\frac{332}{1996}=\frac{83}{499}$. The answer is then $\displaystyle 499+83=582$

See also

1997 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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